What is the integral of (x-1)/(x^2-2x+2) ?

The integral of (x-1)/(x^2-2x+2) is 1/2 ln|x^2-2x+2|+C

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integral of (x-1)/(x^2-2x+2)

\int \frac{x - 1}{x ^{2} - 2 x + 2} d x

\frac{1}{2} ln x^{2} - 2 x + 2 + C

Solution steps

\int \frac{x - 1}{x ^{2} - 2 x + 2} d x

Expand

\frac{x - 1}{x ^{2} - 2 x + 2} : \frac{x}{x ^{2} - 2 x + 2} - \frac{1}{x ^{2} - 2 x + 2}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x}{x ^{2} - 2 x + 2} d x - \int \frac{1}{x ^{2} - 2 x + 2} d x

\int \frac{x}{x ^{2} - 2 x + 2} d x = \frac{1}{2} ln x^{2} - 2 x + 2 + arctan (x - 1)

\int \frac{1}{x ^{2} - 2 x + 2} d x = arctan (x - 1)

= \frac{1}{2} ln x^{2} - 2 x + 2 + arctan (x - 1) - arctan (x - 1)

Simplify

= \frac{1}{2} ln x^{2} - 2 x + 2

Add a constant to the solution

= \frac{1}{2} ln x^{2} - 2 x + 2 + C

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x \to 3 - lim (f (x) x)

\frac{\partial}{\partial x} (ln (x + c t))

What is the integral of (x-1)/(x^2-2x+2) ?
The integral of (x-1)/(x^2-2x+2) is 1/2 ln|x^2-2x+2|+C